This group of order <math>p^3</math> can also be described as a semidirect product of the [[elementary abelian group of prime-square order|elementary abelian group of order]] <math>p^2</math> by the [[group of prime order|cyclic group of order]] <math>p</math>, where the generator of the cyclic group of order <math>p</math> acts via the automorphism:

+

This group of order <math>p^3</math> can also be described as a semidirect product of the [[elementary abelian group of prime-square order|elementary abelian group of order]] <math>p^2</math> by the [[group of prime order|cyclic group of order]] <math>p</math>, with the following action. Denote the base of the semidirect product as ordered pairs of elements from <math>\mathbb{Z}/p\mathbb{Z}</math>. The action of the generator of the acting group is as follows:

−

<math>(a,b) \mapsto (a,a+b)</math>

+

<math>(\alpha,\beta) \mapsto (\alpha,\alpha+\beta)</math>

−

In this case, for instance, we can take the subgroup with <math>a_{12} = 0</math> as the elementary abelian subgroup of order <math>p^2</math> and the subgroup with <math>a_{23} = a_{13} = 0</math> as the cyclic subgroup of order <math>p</math>.

+

In this case, for instance, we can take the subgroup with <math>a_{12} = 0</math> as the elementary abelian subgroup of order <math>p^2</math>, i.e., the elementary abelian subgroup of order <math>p^2</math> is the subgroup:

Definition by presentation

These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators correspond to matrices:

As a semidirect product

This group of order can also be described as a semidirect product of the elementary abelian group of order by the cyclic group of order, with the following action. Denote the base of the semidirect product as ordered pairs of elements from . The action of the generator of the acting group is as follows:

In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order , i.e., the elementary abelian subgroup of order is the subgroup:

The acting subgroup of order can be taken as the subgroup with , i.e., the subgroup:

In coordinate form

We may define the group as set of triples over the prime field,
with the multiplication law given by:

.

The matrix corresponding to triple is:

Families

These groups fall in the more general family of unitriangular matrix groups. The unitriangular matrix group can be described as the group of unipotent upper-triangular matrices in , which is also a -Sylow subgroup of the general linear group. This further can be generalized to where is the power of a prime . is the -Sylow subgroup of .

As an extraspecial group

where the argument indicates that it is the extraspecial group of exponent . For instance, for :

ExtraspecialGroup(5^3,5)

Endomorphisms

Automorphisms

The automorphisms essentially permute the subgroups of order containing the center, while leaving the center itself unmoved.

Related groups

For any prime , there are (up to isomorphism) two non-abelian groups of order . One of them is this, and the other is the semidirect product of the cyclic group of order by a group of order acting by power maps (with the generator corresponding to multiplication by ).